Cyclic graphs are a particular class of regular graphs: they preserve their topology when transformed by a group of symmetries which cyclically takes any one node and maps it into another. Namely, a cyclic permutation is such that \(\pi [\lbrace 1,2,\ldots ,N-1,N\rbrace ] = \lbrace 2,\ldots ,N-1,N,1\rbrace \)  (hence, \(\pi \circ \pi \circ \cdots \circ \pi = \pi ^N = \mathbf {I}\) ), and cyclic graphs are graphs that preserve their local and global topological properties under groups of permutations, \(\lbrace \pi ,\pi ^2,\ldots ,\pi ^N\rbrace \) . This implies that cyclic graphs contain all their connectivity information in any given row of \(\mathbf {L}\)  (or \(\mathbf {A}\) ) and have analytical expressions for their eigenvalues and eigenvectors based on a Fourier basis [1]}, [2]}, [3]}. Hence, we will focus on the first row of \(\mathbf {L}\) , \(\lbrace L_{1,j}\rbrace _{j=1}^{N} = \lbrace k,\,- A_{1,2},\dots ,-A_{1,N}\rbrace \) , and the eigenvalues can be expressed in terms of \(\lbrace L_{1,j}\rbrace \)  as
\(\lambda _n = \sum _{j=1}^{N} L_{1,j} \cos \!\left[\frac{2 \pi n}{N}(j-1)\right] = k - \sum _{j=2}^{N} A_{1,j} \cos \!\left[\frac{2 \pi n}{N}(j-1)\right].\) 
